On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

TL;DR

This paper investigates the non-vanishing of Fourier coefficients of cusp forms of higher weight, establishing conditions under which these coefficients are non-zero in short intervals, and constructing examples with specific non-vanishing properties.

Contribution

It demonstrates non-vanishing of Fourier coefficients in short intervals for cusp forms near elliptic curves with certain isogenies, and constructs explicit non-CM cusp forms with bounded non-zero coefficients.

Findings

Fourier coefficients are non-zero in short intervals for forms close to specific elliptic curves.

Constructs non-CM cusp forms with Fourier coefficients bounded by n^{1/4}.

Provides conditions linking modular forms and elliptic curve isogenies.

Abstract

We show that if a modular cuspidal eigenform $f$ of weight $2k$ is $2$-adically close to an elliptic curve $E/\mathbb{Q}$, which has a cyclic rational $4$-isogeny, then $n$-th Fourier coefficient of $f$ is non-zero in the short interval $(X, X + cX^{\frac{1}{4}})$ for all $X \gg 0$ and for some $c > 0$. We use this fact to produce non-CM cuspidal eigenforms $f$ of level $N>1$ and weight $k > 2$ such that $i_f(n) \ll n^{\frac{1}{4}}$ for all $n \gg 0$.